Riesz mean

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In mathematics, the Riesz mean is a certain mean of an arithmetic series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean^[1]^[2]. The Riesz mean should not be confused with the Bochner-Riesz mean or the Strong-Riesz mean.

[edit] Definition

Given a series ${s n}$ , the Riesz mean of the series is defined by

$s^\delta(\lambda) = \sum_{n\le \lambda} \left(1-\frac{n}{\lambda}\right)^\delta s_n$

Sometimes, a generalized Riesz mean is defined as

$R_n = \frac{1}{\lambda_n} \sum_{k=0}^n (\lambda_k-\lambda_{k-1})^\delta s_k$

Here, the $λ n$ are sequence with $\lambda_n\to\infty$ and with $\lambda_{n+1}/\lambda_n\to 1$ as $n\to\infty$ . Other than this, the $λ n$ are otherwise taken as arbitrary.

Riesz means are often used to explore the summability of sequences; typical summability theorems discuss the case of $s_n = \sum_{k=0}^n a_n$ for some sequence ${a n}$ . Typically, a sequence is summable when the limit $\lim_{n\to\infty} R_n$ exists, or the limit $\lim_{\delta\to 1,\lambda\to\infty}s^\delta(\lambda)$ exists, although the precise summability theorems in question often impose additional conditions.

[edit] Special cases

Let $a n = 1$ for all $n$ . Then

$\sum_{n\le \lambda} \left(1-\frac{n}{\lambda}\right)^\delta = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{\Gamma(1+\delta)\Gamma(s)}{\Gamma(1+\delta+s)} \zeta(s) \lambda^s ds = \frac{\lambda}{1+\delta} + \sum_n b_n \lambda^{-n}$

Here, one must take $c > 1$ ; $Γ(s)$ is the Gamma function and $ζ(s)$ is the Riemann zeta function. The power series

∑	b_nλ ^{− n}
n

can be shown to be convergent for $λ > 1$ . Note that the integral is of the form of an inverse Mellin transform.

Another interesting case connected with number theory arises by taking $a n = Λ(n)$ where $Λ(n)$ is the Von Mangoldt function. Then

$\sum_{n\le \lambda} \left(1-\frac{n}{\lambda}\right)^\delta \Lambda(n) = - \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{\Gamma(1+\delta)\Gamma(s)}{\Gamma(1+\delta+s)} \frac{\zeta^\prime(s)}{\zeta(s)} \lambda^s ds = \frac{\lambda}{1+\delta} + \sum_\rho \frac {\Gamma(1+\delta)\Gamma(\rho)}{\Gamma(1+\delta+\rho)} +\sum_n c_n \lambda^{-n}$

Again, one must take $c > 1$ . The sum over $ρ$ is the sum over the zeroes of the Riemann zeta function, and

∑	c_nλ ^{− n}
n

is convergent for $λ > 1$ .

The integrals that occur here are similar to the Nörlund-Rice integral; very roughly, they can be connected to that integral via Perron's formula.

[edit] References

^ M. Riesz, Comptes Rendus, 12 June 1911
^ G.H. Hardy and J.E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Mathematica, 41(1916) pp.119-196.

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Categories: Means | Zeta and L-functions

Riesz mean

From Wikipedia, the free encyclopedia

[edit] Definition

[edit] Special cases

[edit] References

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